The present invention relates to the field of spectral analysis of digitized signals. More particularly, but without limitation thereto, the present invention is directed to a method of detecting non-Gaussian signals in the presence of Gaussian noise using complex cumulants.
Spectrum analysis is among the most commonly used methods to study probabilistic wave forms. Spectrum analysis usually consists of a plot of the second moment of the Fourier transform coefficients vs. frequency of a signal waveform. This type of plot is a measure of the total power or energy of the waveform, but does not take into account the waveform's inherent statistics. Often, the waveform of interest is masked by a significant amount of Gaussian noise. This has stimulated a search for analysis methods which are less sensitive to Gaussian noise, and which emphasize the statistics of the non-Gaussian components of the waveform of interest. It is generally regarded that one of the best approaches is to use techniques designed to discriminate between statistical distributions. If the signal has a different distribution than the noise, then one needs only to look for changes in the statistical distribution. Currently, one of the most useful distribution tests is to compute the cumulants. Cumulants are functions of the moments of the probability distribution function. They have several properties useful for detecting changes of statistical distribution, one of which is that all cumulants above second order are zero for Gaussian signals. Since the power estimates for spectrum analysis may be regarded as second order cumulants, cumulants are a promising candidate for distribution analysis of power spectra.
Unfortunately, there are several impediments to the use of cumulants for signal detection. For example, it has not been clear which cumulants to use. Most research has concentrated on third order cumulants of broadband signals using real variables. Since these signals are relatively weak in broadband power compared to background noise, the cumulants are often Fourier transformed to produce "bi-spectra". However, this leads to a complicated two-dimensional search space in which it is not clear where to look for the signal. Also, it is not clear that a particular Fourier transform will concentrate the signal power at one point so that the signal may be identified. Worse, odd order cumulants are sensitive primarily to asymmetric features of the wave forms. Asymmetric acoustic signals exist, but they tend to be associated with large displacement sources, which may exclude many signals of interest. If real variable cumulants are to be used, they would probably have to be fourth order cumulants, which would further increase the dimension of the search space.
Another difficulty lies in that it has not been clear how to threshold the cumulant estimates. For example, suppose that in the case of noise only, the fourth order cumulant should be zero, but the estimate equals 100. There is no clear theory that says that a value of 100 is small and therefore not significant, or that it is large and almost certainly indicates the presence of a signal.
Yet another problem is that a signal filtered to improve the signal-to-noise ratio often takes the form of Fourier coefficients, which are complex numbers. Most statistical theory, however, is worked out for real numbers. Ordinarily, functions of real variables can be extended as analytic functions of complex variables in a straightforward manner, but probability density functions are not analytic. The appropriate ideas for handling complex random variables are poorly formulated and not widely known. The generalization of most statistical approaches to complex numbers involves subtle difficulties which are not well-known. Unless one is very conscious of the fact that the data is in complex form, the existing literature and practice tend to steer one toward real variables. It is commonly believed that the filtering necessary to observe narrow band signals will, by the central limit theorem, render them Gaussian. In fact, however, the operation of AM radio demonstrates that this does not happen. An AM radio receiver heterodynes a waveform and low pass filters the product to generate the desired signal. A Fourier transform may also be considered as a heterodyne followed by a low pass filter. The filtering operation represented by integration forms a sum of many samples of the heterodyne waveform. The summation is often assumed to produce a Gaussian waveform according to the central limit theorem. However, the filtered signal bandwidth of the AM radio receiver is only a small fraction of the received signal bandwidth, thus the number of samples effectively summed is at least as large as most Fourier transforms. Yet the filtered signal emerges from the AM radio receiver with all of its non-Gaussian features intact.
Methods for using cumulants to analyze random time series have focused on forming time domain cumulant functions, then Fourier transforming these functions. Such methods result in what is often called "high order spectra", "bi-spectra", etc. This approach has had limited success because it is computationally intensive. The data is multidimensional and therefore difficult to display, and most of the Fourier transform moments computed are of little interest.
A method for discriminating against non-Gaussian interference to detect Gaussian signals is described in U.S. Pat. No. 4,530,076 to Dwyer, Jul. 16, 1985. Although non-Gaussian signals may be suppressed, no attempt is made to preserve the statistics of the non-Gaussian components in a way which would lend itself to detecting and classifying such signals. The novelty of Dwyer's method is the use of Gaussian kurtosis to form quantile estimates.
U.S. Pat. No. 5,091,890 to Higbie, May 21, 1991 discloses a method for parameter estimation using higher order moments or cumulants in the time domain.
U.S. Pat. No. 5,283,813 to Shalvi et al, Feb. 1, 1994 describes a technique for dealing with effects of a filter using second order cumulants. In this instance, the cumulants are parameters in a system of linear equations rather than variables.
U.S. Pat. No. 5,315,532 to Comon, May 24, 1994 describes a spatial processing technique similar to some beamforming methods. Second order power estimates are formed from cumulants.
U.S. Pat. No. 5,343,404 to Girgis, Aug. 30, 1994 describes a method for analyzing harmonic signals consisting of known sets of harmonics.
While these applications of spectral analysis are some indication of the interest in cumulants, a need still exists for successfully applying complex cumulants to a method of analyzing non-Gaussian signals in the presence of Gaussian noise.